Special values of automorphic L-functions

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Special values of automorphic L-functions A. Raghuram Indian Institute of Science Education and Research, Pune, India IAS, Princeton, USA

April 26, 2018

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions of modular forms Theorem (Shimura) Let ϕ ∈ Sk (N, ω)prim . There exists u ± (ϕ) ∈ C∗ (the periods of ϕ) such that for any integer m with 1 ≤ m ≤ k − 1, and any Dirichlet character χ we have Lf (m, ϕ, χ) ∼ (2πi)m γ(χ)u ± (ϕ) where χ(−1) = ±(−1)m . ∼ means up to an element of the number field Q(ϕ, χ) := Q(an (ϕ), values of χ). A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions of modular forms Theorem (Shimura) Let ϕ ∈ Sk (N, ω)prim . There exists u ± (ϕ) ∈ C∗ (the periods of ϕ) such that for any integer m with 1 ≤ m ≤ k − 1, and any Dirichlet character χ we have Lf (m, ϕ, χ) ∼ (2πi)m γ(χ)u ± (ϕ) where χ(−1) = ±(−1)m . ∼ means up to an element of the number field Q(ϕ, χ) := Q(an (ϕ), values of χ). A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions of modular forms Theorem (Shimura) Let ϕ ∈ Sk (N, ω)prim . There exists u ± (ϕ) ∈ C∗ (the periods of ϕ) such that for any integer m with 1 ≤ m ≤ k − 1, and any Dirichlet character χ we have Lf (m, ϕ, χ) ∼ (2πi)m γ(χ)u ± (ϕ) where χ(−1) = ±(−1)m . ∼ means up to an element of the number field Q(ϕ, χ) := Q(an (ϕ), values of χ). A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions of modular forms Theorem (Shimura) Let ϕ ∈ Sk (N, ω)prim . There exists u ± (ϕ) ∈ C∗ (the periods of ϕ) such that for any integer m with 1 ≤ m ≤ k − 1, and any Dirichlet character χ we have Lf (m, ϕ, χ) ∼ (2πi)m γ(χ)u ± (ϕ) where χ(−1) = ±(−1)m . ∼ means up to an element of the number field Q(ϕ, χ) := Q(an (ϕ), values of χ). A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions of modular forms Theorem (Shimura) Let ϕ ∈ Sk (N, ω)prim . There exists u ± (ϕ) ∈ C∗ (the periods of ϕ) such that for any integer m with 1 ≤ m ≤ k − 1, and any Dirichlet character χ we have Lf (m, ϕ, χ) ∼ (2πi)m γ(χ)u ± (ϕ) where χ(−1) = ±(−1)m . ∼ means up to an element of the number field Q(ϕ, χ) := Q(an (ϕ), values of χ). A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions of modular forms Theorem (Shimura) Let ϕ ∈ Sk (N, ω)prim . There exists u ± (ϕ) ∈ C∗ (the periods of ϕ) such that for any integer m with 1 ≤ m ≤ k − 1, and any Dirichlet character χ we have Lf (m, ϕ, χ) ∼ (2πi)m γ(χ)u ± (ϕ) where χ(−1) = ±(−1)m . ∼ means up to an element of the number field Q(ϕ, χ) := Q(an (ϕ), values of χ). A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions of modular forms Theorem (Shimura) Let ϕ ∈ Sk (N, ω)prim . There exists u ± (ϕ) ∈ C∗ (the periods of ϕ) such that for any integer m with 1 ≤ m ≤ k − 1, and any Dirichlet character χ we have Lf (m, ϕ, χ) ∼ (2πi)m γ(χ)u ± (ϕ) where χ(−1) = ±(−1)m . ∼ means up to an element of the number field Q(ϕ, χ) := Q(an (ϕ), values of χ). A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Ratio of successive critical values Corollary (to Shimura’s theorem) Let ϕ ∈ Sk (N, ω)prim . For any integer m with 1 ≤ m ≤ k − 2, and any Dirichlet character χ we have Lf (m, ϕ, χ) u ± (ϕ) ∼ (2πi)−1 ∓ Lf (m + 1, ϕ, χ) u (ϕ) Corollary (Restated...) The quantity 1 Ω(ϕ)χ(−1)(−1)m

L(m, ϕ, χ) L(m + 1, ϕ, χ)

¯ is algebraic and Gal(Q/Q)-equivariant. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Ratio of successive critical values Corollary (to Shimura’s theorem) Let ϕ ∈ Sk (N, ω)prim . For any integer m with 1 ≤ m ≤ k − 2, and any Dirichlet character χ we have u ± (ϕ) Lf (m, ϕ, χ) ∼ (2πi)−1 ∓ Lf (m + 1, ϕ, χ) u (ϕ) Corollary (Restated...) The quantity 1 Ω(ϕ)χ(−1)(−1)m

L(m, ϕ, χ) L(m + 1, ϕ, χ)

¯ is algebraic and Gal(Q/Q)-equivariant. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Ratio of successive critical values Corollary (to Shimura’s theorem) Let ϕ ∈ Sk (N, ω)prim . For any integer m with 1 ≤ m ≤ k − 2, and any Dirichlet character χ we have Lf (m, ϕ, χ) u ± (ϕ) ∼ (2πi)−1 ∓ Lf (m + 1, ϕ, χ) u (ϕ) Corollary (Restated...) The quantity 1 Ω(ϕ)χ(−1)(−1)m

L(m, ϕ, χ) L(m + 1, ϕ, χ)

¯ is algebraic and Gal(Q/Q)-equivariant. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Ratio of successive critical values Corollary (to Shimura’s theorem) Let ϕ ∈ Sk (N, ω)prim . For any integer m with 1 ≤ m ≤ k − 2, and any Dirichlet character χ we have Lf (m, ϕ, χ) u ± (ϕ) ∼ (2πi)−1 ∓ Lf (m + 1, ϕ, χ) u (ϕ) Corollary (Restated...) The quantity 1 Ω(ϕ)χ(−1)(−1)m

L(m, ϕ, χ) L(m + 1, ϕ, χ)

¯ is algebraic and Gal(Q/Q)-equivariant. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Ratio of successive critical values Corollary (to Shimura’s theorem) Let ϕ ∈ Sk (N, ω)prim . For any integer m with 1 ≤ m ≤ k − 2, and any Dirichlet character χ we have Lf (m, ϕ, χ) u ± (ϕ) ∼ (2πi)−1 ∓ Lf (m + 1, ϕ, χ) u (ϕ) Corollary (Restated...) The quantity 1 Ω(ϕ)χ(−1)(−1)m

L(m, ϕ, χ) L(m + 1, ϕ, χ)

¯ is algebraic and Gal(Q/Q)-equivariant. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Ratio of successive critical values Corollary (to Shimura’s theorem) Let ϕ ∈ Sk (N, ω)prim . For any integer m with 1 ≤ m ≤ k − 2, and any Dirichlet character χ we have Lf (m, ϕ, χ) u ± (ϕ) ∼ (2πi)−1 ∓ Lf (m + 1, ϕ, χ) u (ϕ) Corollary (Restated...) The quantity 1 Ω(ϕ)χ(−1)(−1)m

L(m, ϕ, χ) L(m + 1, ϕ, χ)

¯ is algebraic and Gal(Q/Q)-equivariant. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Ratio of successive critical values Corollary (to Shimura’s theorem) Let ϕ ∈ Sk (N, ω)prim . For any integer m with 1 ≤ m ≤ k − 2, and any Dirichlet character χ we have Lf (m, ϕ, χ) u ± (ϕ) ∼ (2πi)−1 ∓ Lf (m + 1, ϕ, χ) u (ϕ) Corollary (Restated...) The quantity 1 Ω(ϕ)χ(−1)(−1)m

L(m, ϕ, χ) L(m + 1, ϕ, χ)

¯ is algebraic and Gal(Q/Q)-equivariant. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions of modular forms

Theorem (Shimura) Let ϕ ∈ Sk (N, ω)prim . Let ξ ∈ Sl (N, ψ)prim . Suppose l 6= k and suppose m and m + 1 are critical for the degree-4 Rankin-Selberg L-function, then L(m, ϕ × ξ) ∼ L(m + 1, ϕ × ξ)

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions of modular forms

Theorem (Shimura) Let ϕ ∈ Sk (N, ω)prim . Let ξ ∈ Sl (N, ψ)prim . Suppose l 6= k and suppose m and m + 1 are critical for the degree-4 Rankin-Selberg L-function, then L(m, ϕ × ξ) ∼ L(m + 1, ϕ × ξ)

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions of modular forms

Theorem (Shimura) Let ϕ ∈ Sk (N, ω)prim . Let ξ ∈ Sl (N, ψ)prim . Suppose l 6= k and suppose m and m + 1 are critical for the degree-4 Rankin-Selberg L-function, then L(m, ϕ × ξ) ∼ L(m + 1, ϕ × ξ)

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions of modular forms

Theorem (Shimura) Let ϕ ∈ Sk (N, ω)prim . Let ξ ∈ Sl (N, ψ)prim . Suppose l 6= k and suppose m and m + 1 are critical for the degree-4 Rankin-Selberg L-function, then L(m, ϕ × ξ) ∼ L(m + 1, ϕ × ξ)

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Rankin–Selberg L-functions (with Günter Harder) Theorem (Harder + R.) Let Gn = GLn /F , with F a totally real field. Let π ∈ Coh(Gn , µ), and π 0 ∈ Coh(Gn0 , µ0 ). Assume that n is even and n0 is odd. Put N = n + n0 . There exists a nonzero complex number Ω(π) depending only on π such that if a combinatorial condition C(µ, µ0 ) holds then L(−N/2, π × π 0 ) 0 ∼Q(π,π0 ) Ω(π) 0 L(1 − N/2, π × π ) for a sign 0 depending only on π 0 .

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Rankin–Selberg L-functions (with Günter Harder) Theorem (Harder + R.) Let Gn = GLn /F , with F a totally real field. Let π ∈ Coh(Gn , µ), and π 0 ∈ Coh(Gn0 , µ0 ). Assume that n is even and n0 is odd. Put N = n + n0 . There exists a nonzero complex number Ω(π) depending only on π such that if a combinatorial condition C(µ, µ0 ) holds then L(−N/2, π × π 0 ) 0 ∼Q(π,π0 ) Ω(π) 0 L(1 − N/2, π × π ) for a sign 0 depending only on π 0 .

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Rankin–Selberg L-functions (with Günter Harder) Theorem (Harder + R.) Let Gn = GLn /F , with F a totally real field. Let π ∈ Coh(Gn , µ), and π 0 ∈ Coh(Gn0 , µ0 ). Assume that n is even and n0 is odd. Put N = n + n0 . There exists a nonzero complex number Ω(π) depending only on π such that if a combinatorial condition C(µ, µ0 ) holds then L(−N/2, π × π 0 ) 0 ∼Q(π,π0 ) Ω(π) 0 L(1 − N/2, π × π ) for a sign 0 depending only on π 0 .

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Rankin–Selberg L-functions (with Günter Harder) Theorem (Harder + R.) Let Gn = GLn /F , with F a totally real field. Let π ∈ Coh(Gn , µ), and π 0 ∈ Coh(Gn0 , µ0 ). Assume that n is even and n0 is odd. Put N = n + n0 . There exists a nonzero complex number Ω(π) depending only on π such that if a combinatorial condition C(µ, µ0 ) holds then L(−N/2, π × π 0 ) 0 ∼Q(π,π0 ) Ω(π) 0 L(1 − N/2, π × π ) for a sign 0 depending only on π 0 .

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Rankin–Selberg L-functions (with Günter Harder) Theorem (Harder + R.) Let Gn = GLn /F , with F a totally real field. Let π ∈ Coh(Gn , µ), and π 0 ∈ Coh(Gn0 , µ0 ). Assume that n is even and n0 is odd. Put N = n + n0 . There exists a nonzero complex number Ω(π) depending only on π such that if a combinatorial condition C(µ, µ0 ) holds then L(−N/2, π × π 0 ) 0 ∼Q(π,π0 ) Ω(π) 0 L(1 − N/2, π × π ) for a sign 0 depending only on π 0 .

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Rankin–Selberg L-functions (with Günter Harder) Theorem (Harder + R.) Let Gn = GLn /F , with F a totally real field. Let π ∈ Coh(Gn , µ), and π 0 ∈ Coh(Gn0 , µ0 ). Assume that n is even and n0 is odd. Put N = n + n0 . There exists a nonzero complex number Ω(π) depending only on π such that if a combinatorial condition C(µ, µ0 ) holds then L(−N/2, π × π 0 ) 0 ∼Q(π,π0 ) Ω(π) 0 L(1 − N/2, π × π ) for a sign 0 depending only on π 0 .

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Rankin–Selberg L-functions (with Günter Harder) Theorem (Harder + R.) Let Gn = GLn /F , with F a totally real field. Let π ∈ Coh(Gn , µ), and π 0 ∈ Coh(Gn0 , µ0 ). Assume that n is even and n0 is odd. Put N = n + n0 . There exists a nonzero complex number Ω(π) depending only on π such that if a combinatorial condition C(µ, µ0 ) holds then L(−N/2, π × π 0 ) 0 ∼Q(π,π0 ) Ω(π) 0 L(1 − N/2, π × π ) for a sign 0 depending only on π 0 .

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

The combinatorial lemma C(µ, µ0 ) The following three conditions on the weights µ and µ0 are equivalent: 1

−N/2 and 1 − N/2 are critical for L(s, π × π 0 ).

2

C(µ, µ0 ) is a condition of the form: abelian − width(µ, µ0 ) ≤ cuspidal − width(µ, µ0 ).

3

There exists w ∈ W P , with l(w) = nn0 /2 and w −1 · (µ + µ0 ) is dominant.

If π 7→ π ⊗ | |r then µ 7→ µ − r for r ∈ Z. Do this as long as C(µ − r , µ0 ) holds. This gives us a rationality theorem for every successive pair of critical values; no more and no less! A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

The combinatorial lemma C(µ, µ0 ) The following three conditions on the weights µ and µ0 are equivalent: 1

−N/2 and 1 − N/2 are critical for L(s, π × π 0 ).

2

C(µ, µ0 ) is a condition of the form: abelian − width(µ, µ0 ) ≤ cuspidal − width(µ, µ0 ).

3

There exists w ∈ W P , with l(w) = nn0 /2 and w −1 · (µ + µ0 ) is dominant.

If π 7→ π ⊗ | |r then µ 7→ µ − r for r ∈ Z. Do this as long as C(µ − r , µ0 ) holds. This gives us a rationality theorem for every successive pair of critical values; no more and no less! A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

The combinatorial lemma C(µ, µ0 ) The following three conditions on the weights µ and µ0 are equivalent: 1

−N/2 and 1 − N/2 are critical for L(s, π × π 0 ).

2

C(µ, µ0 ) is a condition of the form: abelian − width(µ, µ0 ) ≤ cuspidal − width(µ, µ0 ).

3

There exists w ∈ W P , with l(w) = nn0 /2 and w −1 · (µ + µ0 ) is dominant.

If π 7→ π ⊗ | |r then µ 7→ µ − r for r ∈ Z. Do this as long as C(µ − r , µ0 ) holds. This gives us a rationality theorem for every successive pair of critical values; no more and no less! A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

The combinatorial lemma C(µ, µ0 ) The following three conditions on the weights µ and µ0 are equivalent: 1

−N/2 and 1 − N/2 are critical for L(s, π × π 0 ).

2

C(µ, µ0 ) is a condition of the form: abelian − width(µ, µ0 ) ≤ cuspidal − width(µ, µ0 ).

3

There exists w ∈ W P , with l(w) = nn0 /2 and w −1 · (µ + µ0 ) is dominant.

If π 7→ π ⊗ | |r then µ 7→ µ − r for r ∈ Z. Do this as long as C(µ − r , µ0 ) holds. This gives us a rationality theorem for every successive pair of critical values; no more and no less! A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Rankin–Selberg L-functions (with Günter Harder)

Theorem (Harder + R.) Let π ∈ Coh(Gn , µ), and π 0 ∈ Coh(Gn0 , µ0 ). If both n and n0 are even, and if m and m + 1 are critical for L(s, π × π 0 ) then L(m, π × π 0 ) ∼ L(m + 1, π × π 0 )

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Rankin–Selberg L-functions (with Günter Harder)

Theorem (Harder + R.) Let π ∈ Coh(Gn , µ), and π 0 ∈ Coh(Gn0 , µ0 ). If both n and n0 are even, and if m and m + 1 are critical for L(s, π × π 0 ) then L(m, π × π 0 ) ∼ L(m + 1, π × π 0 )

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Rankin–Selberg L-functions (with Günter Harder)

Theorem (Harder + R.) Let π ∈ Coh(Gn , µ), and π 0 ∈ Coh(Gn0 , µ0 ). If both n and n0 are even, and if m and m + 1 are critical for L(s, π × π 0 ) then L(m, π × π 0 ) ∼ L(m + 1, π × π 0 )

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Rankin–Selberg L-functions (with Günter Harder)

Theorem (Harder + R.) Let π ∈ Coh(Gn , µ), and π 0 ∈ Coh(Gn0 , µ0 ). If both n and n0 are even, and if m and m + 1 are critical for L(s, π × π 0 ) then L(m, π × π 0 ) ∼ L(m + 1, π × π 0 )

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Rankin–Selberg L-functions (with Günter Harder)

Theorem (Harder + R.) Let π ∈ Coh(Gn , µ), and π 0 ∈ Coh(Gn0 , µ0 ). If both n and n0 are even, and if m and m + 1 are critical for L(s, π × π 0 ) then L(m, π × π 0 ) ∼ L(m + 1, π × π 0 )

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Rankin–Selberg L-functions over a CM field

Theorem ("Theorem") Let π ∈ Coh(Gn , µ), and π 0 ∈ Coh(Gn0 , µ0 ) but now over a CM-field. If m and m + 1 are critical for L(s, π × π 0 ) then L(m, π × π 0 ) ∼ L(m + 1, π × π 0 ) This theorem has a large intersection in its scope with the works of Michael Harris, Harald Grobner and Jie Lin.

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Rankin–Selberg L-functions over a CM field

Theorem ("Theorem") Let π ∈ Coh(Gn , µ), and π 0 ∈ Coh(Gn0 , µ0 ) but now over a CM-field. If m and m + 1 are critical for L(s, π × π 0 ) then L(m, π × π 0 ) ∼ L(m + 1, π × π 0 ) This theorem has a large intersection in its scope with the works of Michael Harris, Harald Grobner and Jie Lin.

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Rankin–Selberg L-functions over a CM field

Theorem ("Theorem") Let π ∈ Coh(Gn , µ), and π 0 ∈ Coh(Gn0 , µ0 ) but now over a CM-field. If m and m + 1 are critical for L(s, π × π 0 ) then L(m, π × π 0 ) ∼ L(m + 1, π × π 0 ) This theorem has a large intersection in its scope with the works of Michael Harris, Harald Grobner and Jie Lin.

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Rankin–Selberg L-functions over a CM field

Theorem ("Theorem") Let π ∈ Coh(Gn , µ), and π 0 ∈ Coh(Gn0 , µ0 ) but now over a CM-field. If m and m + 1 are critical for L(s, π × π 0 ) then L(m, π × π 0 ) ∼ L(m + 1, π × π 0 ) This theorem has a large intersection in its scope with the works of Michael Harris, Harald Grobner and Jie Lin.

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Rankin–Selberg L-functions over a CM field

Theorem ("Theorem") Let π ∈ Coh(Gn , µ), and π 0 ∈ Coh(Gn0 , µ0 ) but now over a CM-field. If m and m + 1 are critical for L(s, π × π 0 ) then L(m, π × π 0 ) ∼ L(m + 1, π × π 0 ) This theorem has a large intersection in its scope with the works of Michael Harris, Harald Grobner and Jie Lin.

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions for orthgonal groups (with C. Bhagwat) Theorem ("Theorem") Let n = 2r ≥ 2 be an even positive integer. Consider SO(n, n)/Q defined so that the subgroup of all upper-triangular matrices is a Borel subgroup. Let µ be a dominant integral weight written as µ = (µ1 ≥ µ2 ≥ · · · ≥ µn−1 ≥ |µn |), with µj ∈ Z. Let σ be a cuspidal automorphic representation of SO(n, n)/Q. Assume: 1

the Arthur parameter Ψσ is cuspidal on GL2n /Q;

2

σ is globally generic;

3

σ∞ |SO(n,n)(R)0 is a discrete series representation with Harish-Chandra parameter µ + ρn .

Let ◦ χ be a finite order character of Q× \A× . ... (cont.) A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions for orthgonal groups (with C. Bhagwat) Theorem ("Theorem") Let n = 2r ≥ 2 be an even positive integer. Consider SO(n, n)/Q defined so that the subgroup of all upper-triangular matrices is a Borel subgroup. Let µ be a dominant integral weight written as µ = (µ1 ≥ µ2 ≥ · · · ≥ µn−1 ≥ |µn |), with µj ∈ Z. Let σ be a cuspidal automorphic representation of SO(n, n)/Q. Assume: 1

the Arthur parameter Ψσ is cuspidal on GL2n /Q;

2

σ is globally generic;

3

σ∞ |SO(n,n)(R)0 is a discrete series representation with Harish-Chandra parameter µ + ρn .

Let ◦ χ be a finite order character of Q× \A× . ... (cont.) A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions for orthgonal groups (with C. Bhagwat) Theorem ("Theorem") Let n = 2r ≥ 2 be an even positive integer. Consider SO(n, n)/Q defined so that the subgroup of all upper-triangular matrices is a Borel subgroup. Let µ be a dominant integral weight written as µ = (µ1 ≥ µ2 ≥ · · · ≥ µn−1 ≥ |µn |), with µj ∈ Z. Let σ be a cuspidal automorphic representation of SO(n, n)/Q. Assume: 1

the Arthur parameter Ψσ is cuspidal on GL2n /Q;

2

σ is globally generic;

3

σ∞ |SO(n,n)(R)0 is a discrete series representation with Harish-Chandra parameter µ + ρn .

Let ◦ χ be a finite order character of Q× \A× . ... (cont.) A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions for orthgonal groups (with C. Bhagwat) Theorem ("Theorem") Let n = 2r ≥ 2 be an even positive integer. Consider SO(n, n)/Q defined so that the subgroup of all upper-triangular matrices is a Borel subgroup. Let µ be a dominant integral weight written as µ = (µ1 ≥ µ2 ≥ · · · ≥ µn−1 ≥ |µn |), with µj ∈ Z. Let σ be a cuspidal automorphic representation of SO(n, n)/Q. Assume: 1

the Arthur parameter Ψσ is cuspidal on GL2n /Q;

2

σ is globally generic;

3

σ∞ |SO(n,n)(R)0 is a discrete series representation with Harish-Chandra parameter µ + ρn .

Let ◦ χ be a finite order character of Q× \A× . ... (cont.) A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions for orthgonal groups (with C. Bhagwat) Theorem ("Theorem") Let n = 2r ≥ 2 be an even positive integer. Consider SO(n, n)/Q defined so that the subgroup of all upper-triangular matrices is a Borel subgroup. Let µ be a dominant integral weight written as µ = (µ1 ≥ µ2 ≥ · · · ≥ µn−1 ≥ |µn |), with µj ∈ Z. Let σ be a cuspidal automorphic representation of SO(n, n)/Q. Assume: 1

the Arthur parameter Ψσ is cuspidal on GL2n /Q;

2

σ is globally generic;

3

σ∞ |SO(n,n)(R)0 is a discrete series representation with Harish-Chandra parameter µ + ρn .

Let ◦ χ be a finite order character of Q× \A× . ... (cont.) A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions for orthgonal groups (with C. Bhagwat) Theorem ("Theorem") Let n = 2r ≥ 2 be an even positive integer. Consider SO(n, n)/Q defined so that the subgroup of all upper-triangular matrices is a Borel subgroup. Let µ be a dominant integral weight written as µ = (µ1 ≥ µ2 ≥ · · · ≥ µn−1 ≥ |µn |), with µj ∈ Z. Let σ be a cuspidal automorphic representation of SO(n, n)/Q. Assume: 1

the Arthur parameter Ψσ is cuspidal on GL2n /Q;

2

σ is globally generic;

3

σ∞ |SO(n,n)(R)0 is a discrete series representation with Harish-Chandra parameter µ + ρn .

Let ◦ χ be a finite order character of Q× \A× . ... (cont.) A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions for orthgonal groups (with C. Bhagwat) Theorem ("Theorem") Let n = 2r ≥ 2 be an even positive integer. Consider SO(n, n)/Q defined so that the subgroup of all upper-triangular matrices is a Borel subgroup. Let µ be a dominant integral weight written as µ = (µ1 ≥ µ2 ≥ · · · ≥ µn−1 ≥ |µn |), with µj ∈ Z. Let σ be a cuspidal automorphic representation of SO(n, n)/Q. Assume: 1

the Arthur parameter Ψσ is cuspidal on GL2n /Q;

2

σ is globally generic;

3

σ∞ |SO(n,n)(R)0 is a discrete series representation with Harish-Chandra parameter µ + ρn .

Let ◦ χ be a finite order character of Q× \A× . ... (cont.) A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions for orthgonal groups (with C. Bhagwat) Theorem ("Theorem") Let n = 2r ≥ 2 be an even positive integer. Consider SO(n, n)/Q defined so that the subgroup of all upper-triangular matrices is a Borel subgroup. Let µ be a dominant integral weight written as µ = (µ1 ≥ µ2 ≥ · · · ≥ µn−1 ≥ |µn |), with µj ∈ Z. Let σ be a cuspidal automorphic representation of SO(n, n)/Q. Assume: 1

the Arthur parameter Ψσ is cuspidal on GL2n /Q;

2

σ is globally generic;

3

σ∞ |SO(n,n)(R)0 is a discrete series representation with Harish-Chandra parameter µ + ρn .

Let ◦ χ be a finite order character of Q× \A× . ... (cont.) A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions for orthgonal groups (with C. Bhagwat) Theorem (cont.) Then the critical set for the degree-2n completed L-function L(s, ◦ χ × σ) is the finite set of contiguous integers {1 − |µn |, 2 − |µn |, . . . , |µn |}. Assume also that |µn | ≥ 1, and suppose m and m + 1 are both critical, then L(m, ◦ χ × σ) ≈ L(m + 1, ◦ χ × σ), where ≈ means up to an element of a number field Q(◦ χ, σ), and furthermore, all the successive ratios are equivariant under Gal(Q/Q). A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions for orthgonal groups (with C. Bhagwat) Theorem (cont.) Then the critical set for the degree-2n completed L-function L(s, ◦ χ × σ) is the finite set of contiguous integers {1 − |µn |, 2 − |µn |, . . . , |µn |}. Assume also that |µn | ≥ 1, and suppose m and m + 1 are both critical, then L(m, ◦ χ × σ) ≈ L(m + 1, ◦ χ × σ), where ≈ means up to an element of a number field Q(◦ χ, σ), and furthermore, all the successive ratios are equivariant under Gal(Q/Q). A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions for orthgonal groups (with C. Bhagwat) Theorem (cont.) Then the critical set for the degree-2n completed L-function L(s, ◦ χ × σ) is the finite set of contiguous integers {1 − |µn |, 2 − |µn |, . . . , |µn |}. Assume also that |µn | ≥ 1, and suppose m and m + 1 are both critical, then L(m, ◦ χ × σ) ≈ L(m + 1, ◦ χ × σ), where ≈ means up to an element of a number field Q(◦ χ, σ), and furthermore, all the successive ratios are equivariant under Gal(Q/Q). A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

L-functions for orthgonal groups (with C. Bhagwat) Theorem (cont.) Then the critical set for the degree-2n completed L-function L(s, ◦ χ × σ) is the finite set of contiguous integers {1 − |µn |, 2 − |µn |, . . . , |µn |}. Assume also that |µn | ≥ 1, and suppose m and m + 1 are both critical, then L(m, ◦ χ × σ) ≈ L(m + 1, ◦ χ × σ), where ≈ means up to an element of a number field Q(◦ χ, σ), and furthermore, all the successive ratios are equivariant under Gal(Q/Q). A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Asai L-functions (with Muthu Krishnamurthy) Theorem ("Theorem") Let E/F be a quadratic extension of totally real fields. Suppose π ∈ Coh(GLn /E, µ). Let χ be a finite order character over F . Then the critical set for degree n2 twisted Asai L-function L(s, π, As± ⊗ χ) is an explicit contiguous string of half-integers determined by µ. Suppose m and m + 1 are both critical for L(s, π, As± ⊗ χ) then we have: L(m, π, As± ⊗ χ) ≈ L(m + 1, π, As± ⊗ χ), where, by ≈, we mean equality up to an element of the rationality field Q(π, χ). Furthermore, the ratio of the two ¯ L-values is algebraic and is Gal(Q/Q)-equivariant. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Asai L-functions (with Muthu Krishnamurthy) Theorem ("Theorem") Let E/F be a quadratic extension of totally real fields. Suppose π ∈ Coh(GLn /E, µ). Let χ be a finite order character over F . Then the critical set for degree n2 twisted Asai L-function L(s, π, As± ⊗ χ) is an explicit contiguous string of half-integers determined by µ. Suppose m and m + 1 are both critical for L(s, π, As± ⊗ χ) then we have: L(m, π, As± ⊗ χ) ≈ L(m + 1, π, As± ⊗ χ), where, by ≈, we mean equality up to an element of the rationality field Q(π, χ). Furthermore, the ratio of the two ¯ L-values is algebraic and is Gal(Q/Q)-equivariant. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Asai L-functions (with Muthu Krishnamurthy) Theorem ("Theorem") Let E/F be a quadratic extension of totally real fields. Suppose π ∈ Coh(GLn /E, µ). Let χ be a finite order character over F . Then the critical set for degree n2 twisted Asai L-function L(s, π, As± ⊗ χ) is an explicit contiguous string of half-integers determined by µ. Suppose m and m + 1 are both critical for L(s, π, As± ⊗ χ) then we have: L(m, π, As± ⊗ χ) ≈ L(m + 1, π, As± ⊗ χ), where, by ≈, we mean equality up to an element of the rationality field Q(π, χ). Furthermore, the ratio of the two ¯ L-values is algebraic and is Gal(Q/Q)-equivariant. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Asai L-functions (with Muthu Krishnamurthy) Theorem ("Theorem") Let E/F be a quadratic extension of totally real fields. Suppose π ∈ Coh(GLn /E, µ). Let χ be a finite order character over F . Then the critical set for degree n2 twisted Asai L-function L(s, π, As± ⊗ χ) is an explicit contiguous string of half-integers determined by µ. Suppose m and m + 1 are both critical for L(s, π, As± ⊗ χ) then we have: L(m, π, As± ⊗ χ) ≈ L(m + 1, π, As± ⊗ χ), where, by ≈, we mean equality up to an element of the rationality field Q(π, χ). Furthermore, the ratio of the two ¯ L-values is algebraic and is Gal(Q/Q)-equivariant. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Asai L-functions (with Muthu Krishnamurthy) Theorem ("Theorem") Let E/F be a quadratic extension of totally real fields. Suppose π ∈ Coh(GLn /E, µ). Let χ be a finite order character over F . Then the critical set for degree n2 twisted Asai L-function L(s, π, As± ⊗ χ) is an explicit contiguous string of half-integers determined by µ. Suppose m and m + 1 are both critical for L(s, π, As± ⊗ χ) then we have: L(m, π, As± ⊗ χ) ≈ L(m + 1, π, As± ⊗ χ), where, by ≈, we mean equality up to an element of the rationality field Q(π, χ). Furthermore, the ratio of the two ¯ L-values is algebraic and is Gal(Q/Q)-equivariant. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Asai L-functions (with Muthu Krishnamurthy) Theorem ("Theorem") Let E/F be a quadratic extension of totally real fields. Suppose π ∈ Coh(GLn /E, µ). Let χ be a finite order character over F . Then the critical set for degree n2 twisted Asai L-function L(s, π, As± ⊗ χ) is an explicit contiguous string of half-integers determined by µ. Suppose m and m + 1 are both critical for L(s, π, As± ⊗ χ) then we have: L(m, π, As± ⊗ χ) ≈ L(m + 1, π, As± ⊗ χ), where, by ≈, we mean equality up to an element of the rationality field Q(π, χ). Furthermore, the ratio of the two ¯ L-values is algebraic and is Gal(Q/Q)-equivariant. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Asai L-functions (with Muthu Krishnamurthy) Theorem ("Theorem") Let E/F be a quadratic extension of totally real fields. Suppose π ∈ Coh(GLn /E, µ). Let χ be a finite order character over F . Then the critical set for degree n2 twisted Asai L-function L(s, π, As± ⊗ χ) is an explicit contiguous string of half-integers determined by µ. Suppose m and m + 1 are both critical for L(s, π, As± ⊗ χ) then we have: L(m, π, As± ⊗ χ) ≈ L(m + 1, π, As± ⊗ χ), where, by ≈, we mean equality up to an element of the rationality field Q(π, χ). Furthermore, the ratio of the two ¯ L-values is algebraic and is Gal(Q/Q)-equivariant. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Asai L-functions (with Muthu Krishnamurthy) Theorem ("Theorem") Let E/F be a quadratic extension of totally real fields. Suppose π ∈ Coh(GLn /E, µ). Let χ be a finite order character over F . Then the critical set for degree n2 twisted Asai L-function L(s, π, As± ⊗ χ) is an explicit contiguous string of half-integers determined by µ. Suppose m and m + 1 are both critical for L(s, π, As± ⊗ χ) then we have: L(m, π, As± ⊗ χ) ≈ L(m + 1, π, As± ⊗ χ), where, by ≈, we mean equality up to an element of the rationality field Q(π, χ). Furthermore, the ratio of the two ¯ L-values is algebraic and is Gal(Q/Q)-equivariant. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Deligne’s Conjecture Let M be a pure motive over Q with coefficients in a number field E. We have the three realizations: 1 Betti realization HB (M) with Hodge decomposition HB (M) ⊗E C = ⊕H p,q . 2 de Rham realization HdR (M) with a Hodge filtration. 3 `-adic realization H` (M) with a Galois action. The comparison isomorphism between HB (M) ⊗E C → HdR (M) ⊗E C gives two periods c ± (M) ∈ (E ⊗ C)× /E × . The Artin L-function attached to the Galois representation on H` (M) is the motivic L-function L(s, M). Lf (m, M) ∼ (2πi)md

± (M)

A. Raghuram

c ± (M),

±1 = (−1)m .

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Deligne’s Conjecture Let M be a pure motive over Q with coefficients in a number field E. We have the three realizations: 1 Betti realization HB (M) with Hodge decomposition HB (M) ⊗E C = ⊕H p,q . 2 de Rham realization HdR (M) with a Hodge filtration. 3 `-adic realization H` (M) with a Galois action. The comparison isomorphism between HB (M) ⊗E C → HdR (M) ⊗E C gives two periods c ± (M) ∈ (E ⊗ C)× /E × . The Artin L-function attached to the Galois representation on H` (M) is the motivic L-function L(s, M). Lf (m, M) ∼ (2πi)md

± (M)

A. Raghuram

c ± (M),

±1 = (−1)m .

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Deligne’s Conjecture Let M be a pure motive over Q with coefficients in a number field E. We have the three realizations: 1 Betti realization HB (M) with Hodge decomposition HB (M) ⊗E C = ⊕H p,q . 2 de Rham realization HdR (M) with a Hodge filtration. 3 `-adic realization H` (M) with a Galois action. The comparison isomorphism between HB (M) ⊗E C → HdR (M) ⊗E C gives two periods c ± (M) ∈ (E ⊗ C)× /E × . The Artin L-function attached to the Galois representation on H` (M) is the motivic L-function L(s, M). Lf (m, M) ∼ (2πi)md

± (M)

A. Raghuram

c ± (M),

±1 = (−1)m .

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Deligne’s Conjecture Let M be a pure motive over Q with coefficients in a number field E. We have the three realizations: 1 Betti realization HB (M) with Hodge decomposition HB (M) ⊗E C = ⊕H p,q . 2 de Rham realization HdR (M) with a Hodge filtration. 3 `-adic realization H` (M) with a Galois action. The comparison isomorphism between HB (M) ⊗E C → HdR (M) ⊗E C gives two periods c ± (M) ∈ (E ⊗ C)× /E × . The Artin L-function attached to the Galois representation on H` (M) is the motivic L-function L(s, M). Lf (m, M) ∼ (2πi)md

± (M)

A. Raghuram

c ± (M),

±1 = (−1)m .

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Deligne’s Conjecture Let M be a pure motive over Q with coefficients in a number field E. We have the three realizations: 1 Betti realization HB (M) with Hodge decomposition HB (M) ⊗E C = ⊕H p,q . 2 de Rham realization HdR (M) with a Hodge filtration. 3 `-adic realization H` (M) with a Galois action. The comparison isomorphism between HB (M) ⊗E C → HdR (M) ⊗E C gives two periods c ± (M) ∈ (E ⊗ C)× /E × . The Artin L-function attached to the Galois representation on H` (M) is the motivic L-function L(s, M). Lf (m, M) ∼ (2πi)md

± (M)

A. Raghuram

c ± (M),

±1 = (−1)m .

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Deligne’s Conjecture Let M be a pure motive over Q with coefficients in a number field E. We have the three realizations: 1 Betti realization HB (M) with Hodge decomposition HB (M) ⊗E C = ⊕H p,q . 2 de Rham realization HdR (M) with a Hodge filtration. 3 `-adic realization H` (M) with a Galois action. The comparison isomorphism between HB (M) ⊗E C → HdR (M) ⊗E C gives two periods c ± (M) ∈ (E ⊗ C)× /E × . The Artin L-function attached to the Galois representation on H` (M) is the motivic L-function L(s, M). Lf (m, M) ∼ (2πi)md

± (M)

A. Raghuram

c ± (M),

±1 = (−1)m .

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Deligne’s Conjecture Let M be a pure motive over Q with coefficients in a number field E. We have the three realizations: 1 Betti realization HB (M) with Hodge decomposition HB (M) ⊗E C = ⊕H p,q . 2 de Rham realization HdR (M) with a Hodge filtration. 3 `-adic realization H` (M) with a Galois action. The comparison isomorphism between HB (M) ⊗E C → HdR (M) ⊗E C gives two periods c ± (M) ∈ (E ⊗ C)× /E × . The Artin L-function attached to the Galois representation on H` (M) is the motivic L-function L(s, M). Lf (m, M) ∼ (2πi)md

± (M)

A. Raghuram

c ± (M),

±1 = (−1)m .

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Deligne’s Conjecture Let M be a pure motive over Q with coefficients in a number field E. We have the three realizations: 1 Betti realization HB (M) with Hodge decomposition HB (M) ⊗E C = ⊕H p,q . 2 de Rham realization HdR (M) with a Hodge filtration. 3 `-adic realization H` (M) with a Galois action. The comparison isomorphism between HB (M) ⊗E C → HdR (M) ⊗E C gives two periods c ± (M) ∈ (E ⊗ C)× /E × . The Artin L-function attached to the Galois representation on H` (M) is the motivic L-function L(s, M). Lf (m, M) ∼ (2πi)md

± (M)

A. Raghuram

c ± (M),

±1 = (−1)m .

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Deligne’s Conjecture Let M be a pure motive over Q with coefficients in a number field E. We have the three realizations: 1 Betti realization HB (M) with Hodge decomposition HB (M) ⊗E C = ⊕H p,q . 2 de Rham realization HdR (M) with a Hodge filtration. 3 `-adic realization H` (M) with a Galois action. The comparison isomorphism between HB (M) ⊗E C → HdR (M) ⊗E C gives two periods c ± (M) ∈ (E ⊗ C)× /E × . The Artin L-function attached to the Galois representation on H` (M) is the motivic L-function L(s, M). Lf (m, M) ∼ (2πi)md

± (M)

A. Raghuram

c ± (M),

±1 = (−1)m .

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Deligne’s Conjecture (cont.)

Studying the arithmetic of L(m, M) L(m + 1, M) involves studying the ratio of periods c + (M)/c − (M).

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Deligne’s Conjecture (cont.)

Studying the arithmetic of L(m, M) L(m + 1, M) involves studying the ratio of periods c + (M)/c − (M).

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Deligne’s Conjecture (cont.)

Studying the arithmetic of L(m, M) L(m + 1, M) involves studying the ratio of periods c + (M)/c − (M).

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Motivic periods (with Pierre Deligne) Theorem ("Theorem") Let M be a pure motive over Q with coefficients in a field E. Suppose S some multilinearalgebraic structure on M. We let G be the structure group of (M, S) defined as: G := {g ∈ GL(M) : gS = S}◦ , Suppose we are given an algebraic representation of (σ, V ) of G defined over E. To this data {M, S, G, (ρ, V )} we can attach a motive MV . Assume that MV has no middle Hodge type. Assume also that the real Frobenius ι of M is an element of G. Decompose V = V + ⊕ V − into the eigenspaces for the action of ρ(ι), i.e., V ± := {v ∈ V : ρ(ι)v = ±v }. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Motivic periods (with Pierre Deligne) Theorem ("Theorem") Let M be a pure motive over Q with coefficients in a field E. Suppose S some multilinearalgebraic structure on M. We let G be the structure group of (M, S) defined as: G := {g ∈ GL(M) : gS = S}◦ , Suppose we are given an algebraic representation of (σ, V ) of G defined over E. To this data {M, S, G, (ρ, V )} we can attach a motive MV . Assume that MV has no middle Hodge type. Assume also that the real Frobenius ι of M is an element of G. Decompose V = V + ⊕ V − into the eigenspaces for the action of ρ(ι), i.e., V ± := {v ∈ V : ρ(ι)v = ±v }. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Motivic periods (with Pierre Deligne) Theorem ("Theorem") Let M be a pure motive over Q with coefficients in a field E. Suppose S some multilinearalgebraic structure on M. We let G be the structure group of (M, S) defined as: G := {g ∈ GL(M) : gS = S}◦ , Suppose we are given an algebraic representation of (σ, V ) of G defined over E. To this data {M, S, G, (ρ, V )} we can attach a motive MV . Assume that MV has no middle Hodge type. Assume also that the real Frobenius ι of M is an element of G. Decompose V = V + ⊕ V − into the eigenspaces for the action of ρ(ι), i.e., V ± := {v ∈ V : ρ(ι)v = ±v }. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Motivic periods (with Pierre Deligne) Theorem ("Theorem") Let M be a pure motive over Q with coefficients in a field E. Suppose S some multilinearalgebraic structure on M. We let G be the structure group of (M, S) defined as: G := {g ∈ GL(M) : gS = S}◦ , Suppose we are given an algebraic representation of (σ, V ) of G defined over E. To this data {M, S, G, (ρ, V )} we can attach a motive MV . Assume that MV has no middle Hodge type. Assume also that the real Frobenius ι of M is an element of G. Decompose V = V + ⊕ V − into the eigenspaces for the action of ρ(ι), i.e., V ± := {v ∈ V : ρ(ι)v = ±v }. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Motivic periods (with Pierre Deligne) Theorem ("Theorem") Let M be a pure motive over Q with coefficients in a field E. Suppose S some multilinearalgebraic structure on M. We let G be the structure group of (M, S) defined as: G := {g ∈ GL(M) : gS = S}◦ , Suppose we are given an algebraic representation of (σ, V ) of G defined over E. To this data {M, S, G, (ρ, V )} we can attach a motive MV . Assume that MV has no middle Hodge type. Assume also that the real Frobenius ι of M is an element of G. Decompose V = V + ⊕ V − into the eigenspaces for the action of ρ(ι), i.e., V ± := {v ∈ V : ρ(ι)v = ±v }. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Motivic periods (with Pierre Deligne) Theorem ("Theorem") Let M be a pure motive over Q with coefficients in a field E. Suppose S some multilinearalgebraic structure on M. We let G be the structure group of (M, S) defined as: G := {g ∈ GL(M) : gS = S}◦ , Suppose we are given an algebraic representation of (σ, V ) of G defined over E. To this data {M, S, G, (ρ, V )} we can attach a motive MV . Assume that MV has no middle Hodge type. Assume also that the real Frobenius ι of M is an element of G. Decompose V = V + ⊕ V − into the eigenspaces for the action of ρ(ι), i.e., V ± := {v ∈ V : ρ(ι)v = ±v }. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Motivic periods (with Pierre Deligne) Theorem ("Theorem") Let M be a pure motive over Q with coefficients in a field E. Suppose S some multilinearalgebraic structure on M. We let G be the structure group of (M, S) defined as: G := {g ∈ GL(M) : gS = S}◦ , Suppose we are given an algebraic representation of (σ, V ) of G defined over E. To this data {M, S, G, (ρ, V )} we can attach a motive MV . Assume that MV has no middle Hodge type. Assume also that the real Frobenius ι of M is an element of G. Decompose V = V + ⊕ V − into the eigenspaces for the action of ρ(ι), i.e., V ± := {v ∈ V : ρ(ι)v = ±v }. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Motivic periods (with Pierre Deligne) Theorem ("Theorem") Let M be a pure motive over Q with coefficients in a field E. Suppose S some multilinearalgebraic structure on M. We let G be the structure group of (M, S) defined as: G := {g ∈ GL(M) : gS = S}◦ , Suppose we are given an algebraic representation of (σ, V ) of G defined over E. To this data {M, S, G, (ρ, V )} we can attach a motive MV . Assume that MV has no middle Hodge type. Assume also that the real Frobenius ι of M is an element of G. Decompose V = V + ⊕ V − into the eigenspaces for the action of ρ(ι), i.e., V ± := {v ∈ V : ρ(ι)v = ±v }. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Motivic periods (with Pierre Deligne) Theorem ("Theorem") Let M be a pure motive over Q with coefficients in a field E. Suppose S some multilinearalgebraic structure on M. We let G be the structure group of (M, S) defined as: G := {g ∈ GL(M) : gS = S}◦ , Suppose we are given an algebraic representation of (σ, V ) of G defined over E. To this data {M, S, G, (ρ, V )} we can attach a motive MV . Assume that MV has no middle Hodge type. Assume also that the real Frobenius ι of M is an element of G. Decompose V = V + ⊕ V − into the eigenspaces for the action of ρ(ι), i.e., V ± := {v ∈ V : ρ(ι)v = ±v }. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Motivic periods (with Pierre Deligne) Theorem (...cont.) Let ZG (ι) := {g ∈ G : gι = ιg} be the centralizer of ι in G. Then ZG (ι) stabilizes V ± under the representation ρ. Define algebraic characters χ± : ZG (ι) → E × by ZG (ι)

/ GL(V ± ) JJJ JJJ det JJJ χ± J% 

E×

If χ+ = χ− then c + (MV ) ∼ c − (MV ).

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Motivic periods (with Pierre Deligne) Theorem (...cont.) Let ZG (ι) := {g ∈ G : gι = ιg} be the centralizer of ι in G. Then ZG (ι) stabilizes V ± under the representation ρ. Define algebraic characters χ± : ZG (ι) → E × by ZG (ι)

/ GL(V ± ) JJJ JJJ det JJJ χ± J% 

E×

If χ+ = χ− then c + (MV ) ∼ c − (MV ).

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Motivic periods (with Pierre Deligne) Theorem (...cont.) Let ZG (ι) := {g ∈ G : gι = ιg} be the centralizer of ι in G. Then ZG (ι) stabilizes V ± under the representation ρ. Define algebraic characters χ± : ZG (ι) → E × by ZG (ι)

/ GL(V ± ) JJJ JJJ det JJJ χ± J% 

E×

If χ+ = χ− then c + (MV ) ∼ c − (MV ).

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Motivic periods (with Pierre Deligne) Theorem (...cont.) Let ZG (ι) := {g ∈ G : gι = ιg} be the centralizer of ι in G. Then ZG (ι) stabilizes V ± under the representation ρ. Define algebraic characters χ± : ZG (ι) → E × by ZG (ι)

/ GL(V ± ) JJJ JJJ det JJJ χ± J% 

E×

If χ+ = χ− then c + (MV ) ∼ c − (MV ).

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Example: Orthogonal motives Let M be a pure motive over Q of rank 2n and purity weight w, and suppose β : M ⊗ M → Q(χ)(−w) is a symmetric nondegenerate morphism of motives. (Such a β gives an orthogonal structure on M.) Assume 1

M has no middle Hodge type, (hence d + = d − = n),

2

n is even, and

3

εβ = 1, i.e., χ(−1) = (−1)w ,

then c + (M) ∼ c − (M). This result, together with Deligne’s conjecture and the Langlands program, implies the results on special values of degree-2n L-functions for SO(n, n).

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Example: Orthogonal motives Let M be a pure motive over Q of rank 2n and purity weight w, and suppose β : M ⊗ M → Q(χ)(−w) is a symmetric nondegenerate morphism of motives. (Such a β gives an orthogonal structure on M.) Assume 1

M has no middle Hodge type, (hence d + = d − = n),

2

n is even, and

3

εβ = 1, i.e., χ(−1) = (−1)w ,

then c + (M) ∼ c − (M). This result, together with Deligne’s conjecture and the Langlands program, implies the results on special values of degree-2n L-functions for SO(n, n).

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Example: Orthogonal motives Let M be a pure motive over Q of rank 2n and purity weight w, and suppose β : M ⊗ M → Q(χ)(−w) is a symmetric nondegenerate morphism of motives. (Such a β gives an orthogonal structure on M.) Assume 1

M has no middle Hodge type, (hence d + = d − = n),

2

n is even, and

3

εβ = 1, i.e., χ(−1) = (−1)w ,

then c + (M) ∼ c − (M). This result, together with Deligne’s conjecture and the Langlands program, implies the results on special values of degree-2n L-functions for SO(n, n).

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Example: Orthogonal motives Let M be a pure motive over Q of rank 2n and purity weight w, and suppose β : M ⊗ M → Q(χ)(−w) is a symmetric nondegenerate morphism of motives. (Such a β gives an orthogonal structure on M.) Assume 1

M has no middle Hodge type, (hence d + = d − = n),

2

n is even, and

3

εβ = 1, i.e., χ(−1) = (−1)w ,

then c + (M) ∼ c − (M). This result, together with Deligne’s conjecture and the Langlands program, implies the results on special values of degree-2n L-functions for SO(n, n).

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Example: Orthogonal motives Let M be a pure motive over Q of rank 2n and purity weight w, and suppose β : M ⊗ M → Q(χ)(−w) is a symmetric nondegenerate morphism of motives. (Such a β gives an orthogonal structure on M.) Assume 1

M has no middle Hodge type, (hence d + = d − = n),

2

n is even, and

3

εβ = 1, i.e., χ(−1) = (−1)w ,

then c + (M) ∼ c − (M). This result, together with Deligne’s conjecture and the Langlands program, implies the results on special values of degree-2n L-functions for SO(n, n).

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Example: Asai motives

Let F /Q be a real quadratic extension. Let M be a pure motive of rank n over F with coefficients in E. Then we have the Asai motives As± (M) both of which are rank n2 -motives over Q with coefficients in E. Assume n is even, and that As± (M) have no middle Hodge type. Then c + (As± (M)) ∼ c − (As± (M)). This result, together with Deligne’s conjecture and the Langlands program, implies the results on special values of degree-n2 Asai L-functions for GL(n)/F .

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Example: Asai motives

Let F /Q be a real quadratic extension. Let M be a pure motive of rank n over F with coefficients in E. Then we have the Asai motives As± (M) both of which are rank n2 -motives over Q with coefficients in E. Assume n is even, and that As± (M) have no middle Hodge type. Then c + (As± (M)) ∼ c − (As± (M)). This result, together with Deligne’s conjecture and the Langlands program, implies the results on special values of degree-n2 Asai L-functions for GL(n)/F .

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Example: Asai motives

Let F /Q be a real quadratic extension. Let M be a pure motive of rank n over F with coefficients in E. Then we have the Asai motives As± (M) both of which are rank n2 -motives over Q with coefficients in E. Assume n is even, and that As± (M) have no middle Hodge type. Then c + (As± (M)) ∼ c − (As± (M)). This result, together with Deligne’s conjecture and the Langlands program, implies the results on special values of degree-n2 Asai L-functions for GL(n)/F .

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Example: Asai motives

Let F /Q be a real quadratic extension. Let M be a pure motive of rank n over F with coefficients in E. Then we have the Asai motives As± (M) both of which are rank n2 -motives over Q with coefficients in E. Assume n is even, and that As± (M) have no middle Hodge type. Then c + (As± (M)) ∼ c − (As± (M)). This result, together with Deligne’s conjecture and the Langlands program, implies the results on special values of degree-n2 Asai L-functions for GL(n)/F .

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Example: Asai motives

Let F /Q be a real quadratic extension. Let M be a pure motive of rank n over F with coefficients in E. Then we have the Asai motives As± (M) both of which are rank n2 -motives over Q with coefficients in E. Assume n is even, and that As± (M) have no middle Hodge type. Then c + (As± (M)) ∼ c − (As± (M)). This result, together with Deligne’s conjecture and the Langlands program, implies the results on special values of degree-n2 Asai L-functions for GL(n)/F .

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Example: Asai motives

Let F /Q be a real quadratic extension. Let M be a pure motive of rank n over F with coefficients in E. Then we have the Asai motives As± (M) both of which are rank n2 -motives over Q with coefficients in E. Assume n is even, and that As± (M) have no middle Hodge type. Then c + (As± (M)) ∼ c − (As± (M)). This result, together with Deligne’s conjecture and the Langlands program, implies the results on special values of degree-n2 Asai L-functions for GL(n)/F .

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Example: Asai motives

Let F /Q be a real quadratic extension. Let M be a pure motive of rank n over F with coefficients in E. Then we have the Asai motives As± (M) both of which are rank n2 -motives over Q with coefficients in E. Assume n is even, and that As± (M) have no middle Hodge type. Then c + (As± (M)) ∼ c − (As± (M)). This result, together with Deligne’s conjecture and the Langlands program, implies the results on special values of degree-n2 Asai L-functions for GL(n)/F .

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Eine kleine Einführung: Eisenstein Kohomologie Let G be a connected reductive group over Q. Let K∞ be the maximal compact subgroup of G∞ = G(R) thickened by the maximal central split torus. Define 0 SKGf := G(Q)\G(A)/K∞ Kf . Let Eλ be a finite-dimensional irreducible representation of G with highest weight λ. It is defined over Q. Let Eλ be the corresponding local system on SKGf . We are interested in the arithmetic information contained in the G(Af ) × π0 (G∞ )-modules H • (S G , Eλ ) := lim H • (SKGf , Eλ ). −→ Kf A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Eine kleine Einführung: Eisenstein Kohomologie Let G be a connected reductive group over Q. Let K∞ be the maximal compact subgroup of G∞ = G(R) thickened by the maximal central split torus. Define 0 SKGf := G(Q)\G(A)/K∞ Kf . Let Eλ be a finite-dimensional irreducible representation of G with highest weight λ. It is defined over Q. Let Eλ be the corresponding local system on SKGf . We are interested in the arithmetic information contained in the G(Af ) × π0 (G∞ )-modules H • (S G , Eλ ) := lim H • (SKGf , Eλ ). −→ Kf A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Eine kleine Einführung: Eisenstein Kohomologie Let G be a connected reductive group over Q. Let K∞ be the maximal compact subgroup of G∞ = G(R) thickened by the maximal central split torus. Define 0 SKGf := G(Q)\G(A)/K∞ Kf . Let Eλ be a finite-dimensional irreducible representation of G with highest weight λ. It is defined over Q. Let Eλ be the corresponding local system on SKGf . We are interested in the arithmetic information contained in the G(Af ) × π0 (G∞ )-modules H • (S G , Eλ ) := lim H • (SKGf , Eλ ). −→ Kf A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Eine kleine Einführung: Eisenstein Kohomologie Let G be a connected reductive group over Q. Let K∞ be the maximal compact subgroup of G∞ = G(R) thickened by the maximal central split torus. Define 0 SKGf := G(Q)\G(A)/K∞ Kf . Let Eλ be a finite-dimensional irreducible representation of G with highest weight λ. It is defined over Q. Let Eλ be the corresponding local system on SKGf . We are interested in the arithmetic information contained in the G(Af ) × π0 (G∞ )-modules H • (S G , Eλ ) := lim H • (SKGf , Eλ ). −→ Kf A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Eine kleine Einführung: Eisenstein Kohomologie Let G be a connected reductive group over Q. Let K∞ be the maximal compact subgroup of G∞ = G(R) thickened by the maximal central split torus. Define 0 SKGf := G(Q)\G(A)/K∞ Kf . Let Eλ be a finite-dimensional irreducible representation of G with highest weight λ. It is defined over Q. Let Eλ be the corresponding local system on SKGf . We are interested in the arithmetic information contained in the G(Af ) × π0 (G∞ )-modules H • (S G , Eλ ) := lim H • (SKGf , Eλ ). −→ Kf A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Eine kleine Einführung: Eisenstein Kohomologie Let G be a connected reductive group over Q. Let K∞ be the maximal compact subgroup of G∞ = G(R) thickened by the maximal central split torus. Define 0 SKGf := G(Q)\G(A)/K∞ Kf . Let Eλ be a finite-dimensional irreducible representation of G with highest weight λ. It is defined over Q. Let Eλ be the corresponding local system on SKGf . We are interested in the arithmetic information contained in the G(Af ) × π0 (G∞ )-modules H • (SKGf , Eλ ). H • (S G , Eλ ) := lim −→ Kf A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Eine kleine Einführung: Eisenstein Kohomologie Let G be a connected reductive group over Q. Let K∞ be the maximal compact subgroup of G∞ = G(R) thickened by the maximal central split torus. Define 0 SKGf := G(Q)\G(A)/K∞ Kf . Let Eλ be a finite-dimensional irreducible representation of G with highest weight λ. It is defined over Q. Let Eλ be the corresponding local system on SKGf . We are interested in the arithmetic information contained in the G(Af ) × π0 (G∞ )-modules H • (SKGf , Eλ ). H • (S G , Eλ ) := lim −→ Kf A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Inner cohomology

Inner cohomology is defined as: H!• (S G , Eλ ) := Image(Hc• (S G , Eλ ) → H • (S G , Eλ )) Inside inner cohomology is a transcendentally defined subspace called cuspidal cohomology. • (..) ⊂ H!• (..) ⊂ H • (..) Hcusp

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Inner cohomology

Inner cohomology is defined as: H!• (S G , Eλ ) := Image(Hc• (S G , Eλ ) → H • (S G , Eλ )) Inside inner cohomology is a transcendentally defined subspace called cuspidal cohomology. • Hcusp (..) ⊂ H!• (..) ⊂ H • (..)

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Inner cohomology

Inner cohomology is defined as: H!• (S G , Eλ ) := Image(Hc• (S G , Eλ ) → H • (S G , Eλ )) Inside inner cohomology is a transcendentally defined subspace called cuspidal cohomology. • Hcusp (..) ⊂ H!• (..) ⊂ H • (..)

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Inner cohomology

Inner cohomology is defined as: H!• (S G , Eλ ) := Image(Hc• (S G , Eλ ) → H • (S G , Eλ )) Inside inner cohomology is a transcendentally defined subspace called cuspidal cohomology. • Hcusp (..) ⊂ H!• (..) ⊂ H • (..)

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Cuspidal cohomology Cuspidal cohomology is defined by the diagram: O

0 / H • (g∞ , K∞ ; C ∞ (G(Q)\G(A)) ⊗ Eλ ) O

• Hcusp (S G , Eλ )

? 0 ∞ / H • (g∞ , K∞ ; Ccusp (G(Q)\G(A)) ⊗ Eλ )

H • (S G , Eλ )

?

Let G = GLn /Q and bn = [n2 /4]. As a G(Af ) × π0 (G∞ )-module we have a multiplicity free decomposition: M bn bn Hcusp (S G , Eλ ) = Hcusp (S G , Eλ )[Πf × ]

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Cuspidal cohomology Cuspidal cohomology is defined by the diagram: O

0 / H • (g∞ , K∞ ; C ∞ (G(Q)\G(A)) ⊗ Eλ ) O

• Hcusp (S G , Eλ )

? 0 ∞ / H • (g∞ , K∞ ; Ccusp (G(Q)\G(A)) ⊗ Eλ )

H • (S G , Eλ )

?

Let G = GLn /Q and bn = [n2 /4]. As a G(Af ) × π0 (G∞ )-module we have a multiplicity free decomposition: M bn bn Hcusp (S G , Eλ ) = Hcusp (S G , Eλ )[Πf × ]

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Cuspidal cohomology Cuspidal cohomology is defined by the diagram: O

0 / H • (g∞ , K∞ ; C ∞ (G(Q)\G(A)) ⊗ Eλ ) O

• Hcusp (S G , Eλ )

? 0 ∞ / H • (g∞ , K∞ ; Ccusp (G(Q)\G(A)) ⊗ Eλ )

H • (S G , Eλ )

?

Let G = GLn /Q and bn = [n2 /4]. As a G(Af ) × π0 (G∞ )-module we have a multiplicity free decomposition: M bn bn Hcusp (S G , Eλ ) = Hcusp (S G , Eλ )[Πf × ]

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Cuspidal cohomology Cuspidal cohomology is defined by the diagram: O

0 / H • (g∞ , K∞ ; C ∞ (G(Q)\G(A)) ⊗ Eλ ) O

• Hcusp (S G , Eλ )

? 0 ∞ / H • (g∞ , K∞ ; Ccusp (G(Q)\G(A)) ⊗ Eλ )

H • (S G , Eλ )

?

Let G = GLn /Q and bn = [n2 /4]. As a G(Af ) × π0 (G∞ )-module we have a multiplicity free decomposition: M bn bn Hcusp (S G , Eλ ) = Hcusp (S G , Eλ )[Πf × ]

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Borel-Serre compactification

¯ be the Borel-Serre For brevity, let M = SKGf , and let M compactification. Here are a few details: ¯ = M ∪ ∂ M, ¯ where the ‘boundary’ ∂ M ¯ is stratified as M ¯ = ∪P ∂P M, the union running over conjugacy classes of ∂M parabolic Q-subgroups P of G. ¯ is a manifold with corners. M ¯ and the inclusion M ,→ M ¯ is a M is the interior of M • • ¯ E). homotopy equivalence, hence, H (M, E) = H (M,

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Borel-Serre compactification

¯ be the Borel-Serre For brevity, let M = SKGf , and let M compactification. Here are a few details: ¯ = M ∪ ∂ M, ¯ where the ‘boundary’ ∂ M ¯ is stratified as M ¯ = ∪P ∂P M, the union running over conjugacy classes of ∂M parabolic Q-subgroups P of G. ¯ is a manifold with corners. M ¯ and the inclusion M ,→ M ¯ is a M is the interior of M • • ¯ E). homotopy equivalence, hence, H (M, E) = H (M,

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Borel-Serre compactification

¯ be the Borel-Serre For brevity, let M = SKGf , and let M compactification. Here are a few details: ¯ = M ∪ ∂ M, ¯ where the ‘boundary’ ∂ M ¯ is stratified as M ¯ = ∪P ∂P M, the union running over conjugacy classes of ∂M parabolic Q-subgroups P of G. ¯ is a manifold with corners. M ¯ and the inclusion M ,→ M ¯ is a M is the interior of M • • ¯ E). homotopy equivalence, hence, H (M, E) = H (M,

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Borel-Serre compactification

¯ be the Borel-Serre For brevity, let M = SKGf , and let M compactification. Here are a few details: ¯ = M ∪ ∂ M, ¯ where the ‘boundary’ ∂ M ¯ is stratified as M ¯ = ∪P ∂P M, the union running over conjugacy classes of ∂M parabolic Q-subgroups P of G. ¯ is a manifold with corners. M ¯ and the inclusion M ,→ M ¯ is a M is the interior of M • • ¯ E). homotopy equivalence, hence, H (M, E) = H (M,

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Borel-Serre compactification

¯ be the Borel-Serre For brevity, let M = SKGf , and let M compactification. Here are a few details: ¯ = M ∪ ∂ M, ¯ where the ‘boundary’ ∂ M ¯ is stratified as M ¯ = ∪P ∂P M, the union running over conjugacy classes of ∂M parabolic Q-subgroups P of G. ¯ is a manifold with corners. M ¯ and the inclusion M ,→ M ¯ is a M is the interior of M • • ¯ E). homotopy equivalence, hence, H (M, E) = H (M,

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

A long exact sequence

A fundamental long exact sequence associated to the pair ¯ ∂ M) ¯ is (M, ι∗

r∗

¯ E) −→ H i+1 (M, E) −→ · · · ¯ E) −→ H i (∂ M, · · · −→ Hci (M, E) −→ H i (M, c

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

What is Eisenstein Cohomology? Eisenstein cohomology gets you back into the manifold SKGf from the boundary ∂SKGf : ···

/ Hci (M, E)

t ¯ E) / H i (M,

Eis∗ r∗

¯ E) / H i (∂ M,

/ ···

Eisenstein cohomology is the image of global cohomology in the cohomology of the boundary i (S G , E) = Image(H i (S G , E) → H i (∂S G , E)). HEis

Eisenstein cohomology consists of cohomology classes represented by cocycles built out of Eisenstein series. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

What is Eisenstein Cohomology? Eisenstein cohomology gets you back into the manifold SKGf from the boundary ∂SKGf : ···

/ Hci (M, E)

t ¯ E) / H i (M,

Eis∗ r∗

¯ E) / H i (∂ M,

/ ···

Eisenstein cohomology is the image of global cohomology in the cohomology of the boundary i (S G , E) = Image(H i (S G , E) → H i (∂S G , E)). HEis

Eisenstein cohomology consists of cohomology classes represented by cocycles built out of Eisenstein series. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

What is Eisenstein Cohomology? Eisenstein cohomology gets you back into the manifold SKGf from the boundary ∂SKGf : ···

/ Hci (M, E)

t ¯ E) / H i (M,

Eis∗ r∗

¯ E) / H i (∂ M,

/ ···

Eisenstein cohomology is the image of global cohomology in the cohomology of the boundary i (S G , E) = Image(H i (S G , E) → H i (∂S G , E)). HEis

Eisenstein cohomology consists of cohomology classes represented by cocycles built out of Eisenstein series. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

What is Eisenstein Cohomology? Eisenstein cohomology gets you back into the manifold SKGf from the boundary ∂SKGf : ···

/ Hci (M, E)

t ¯ E) / H i (M,

Eis∗ r∗

¯ E) / H i (∂ M,

/ ···

Eisenstein cohomology is the image of global cohomology in the cohomology of the boundary i (S G , E) = Image(H i (S G , E) → H i (∂S G , E)). HEis

Eisenstein cohomology consists of cohomology classes represented by cocycles built out of Eisenstein series. A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Cohomology of the (maximal) boundary strata

Let P = MP UP be a maximal proper parabolic subgroup. H • (∂P S G , Eλ )

a

=

=

L

G(A )×π (G

)

∞ • MP , H• (uP , Eλ ))) IndP(Aff )×π00(P∞ ) (H (S

w∈W P

A. Raghuram

a

G(A )×π (G

)

∞ •−l(w) IndP(Aff )×π00(P∞ (S MP , Ew·λ )). ) (H

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Cohomology of the (maximal) boundary strata

Let P = MP UP be a maximal proper parabolic subgroup. H • (∂P S G , Eλ )

a

=

=

L

G(A )×π (G

)

∞ • MP IndP(Aff )×π00(P∞ , H• (uP , Eλ ))) ) (H (S

w∈W P

A. Raghuram

a

G(A )×π (G

)

∞ •−l(w) IndP(Aff )×π00(P∞ (S MP , Ew·λ )). ) (H

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Cohomology of the (maximal) boundary strata

Let P = MP UP be a maximal proper parabolic subgroup. H • (∂P S G , Eλ )

a

=

=

L

G(A )×π (G

)

∞ • MP IndP(Aff )×π00(P∞ , H• (uP , Eλ ))) ) (H (S

w∈W P

A. Raghuram

a

G(A )×π (G

)

∞ •−l(w) IndP(Aff )×π00(P∞ (S MP , Ew·λ )). ) (H

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Gist of it...

So far: Representations induced from cohomological cuspidal representations appear in boundary cohomology. We will see: The Eisenstein classes attached to sections of these induced representations carry arithmetic information about the associated Langlands L-functions.

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Gist of it...

So far: Representations induced from cohomological cuspidal representations appear in boundary cohomology. We will see: The Eisenstein classes attached to sections of these induced representations carry arithmetic information about the associated Langlands L-functions.

A. Raghuram

A classical example Automorphic L-functions Periods of motives Adumbrating the proofs

Gist of it...

So far: Representations induced from cohomological cuspidal representations appear in boundary cohomology. We will see: The Eisenstein classes attached to sections of these induced representations carry arithmetic information about the associated Langlands L-functions.

A. Raghuram